Redundancy — cold standby, hot standby, active redundancy: Foundations

Before you can read the parent note on cold/hot/active redundancy, you need every symbol it throws at you built from nothing. We go one brick at a time. Nothing below assumes you have seen probability, exponentials, or calculus before.

1. Reliability $R (t)$ — the survival chance

The picture: imagine $1000$ identical radios all switched on at the same moment. As hours pass, some die. If after $5$ hours $900$ still work, then $R (5 hours) = 900/1000 = 0.9$ .

Figure — Redundancy — cold standby, hot standby, active redundancy

Why the topic needs it: every claim like "two units are more reliable than one" is a claim about $R (t)$ . Look at the curve in the figure — it starts at $1$ (all alive at $t = 0$ ) and slides toward $0$ . Redundancy is the art of pushing that curve to the right so it stays high for longer.

2. Failure rate $λ$ — how fast things break

The picture: think of $λ$ as the steepness of the death slide. A cheap part has a steep slide (large $λ$ , dies fast); a space-grade part has a gentle slide (tiny $λ$ ).

Why this tool and not another? We use a single constant $λ$ because, for electronics in their useful mid-life, the risk among survivors stays roughly steady — a chip that has lasted 1000 hours is no more likely to die in its next hour than a fresh one. That "memoryless" behaviour is exactly what a constant $λ$ captures, and it is what makes the tidy formulas in the parent note possible.

3. The exponential $e^{- λ t}$ — why survival decays this shape

The picture: it is the death slide of Section 1, written as a formula. See the figure below where a big $λ$ (coral) drops fast and a small $λ$ (mint) drops slowly.

Why the topic needs it: every reliability formula in the parent — $R_{hot}$ , $R_{cold}$ , $R_{TMR}$ — is built by combining copies of $e^{- λ t}$ . Master this one building block and the rest is arithmetic.

Recall Quick check: what is

$R (0)$ for any component? $R (0) = e^{- λ \cdot 0} = e^{0} = 1$ . At time zero everything is alive. ::: $R (0) = 1$

4. "Probability it has failed" — the complement $1 - e^{- λ t}$

Why the topic needs it: hot standby fails only when both units are dead. To compute "both dead" we first need "one is dead", which is exactly this complement. Hold onto the phrase: dead-chance $= 1 - e^{- λ t}$ .

5. Combining independent chances — multiply

The picture: think of a $10 \times 10$ grid of outcomes. If A fails in $30%$ of columns and B fails in $30%$ of rows, the "both fail" corner is $0.3 \times 0.3 = 0.09$ of the whole square.

Why this tool and not addition? We multiply, not add, because we want the overlap ("A dead AND B dead"), not the total ("A dead OR B dead"). This is the single most important move in the hot-standby derivation: system reliability $= 1 - (both dead) = 1 - (1 - e^{- λ t})^{2}$ .

6. MTTF and the integral $\int_{0}^{\infty} R (t) d t$

The picture: in the figure the shaded lavender region is the MTTF. A curve that stays high longer encloses more area — more average life. That is redundancy, geometrically.

The gentler the death slope (small $λ$ ), the longer the average life $1/ λ$ — which matches intuition perfectly.

7. Choosing among outputs — voting and the majority

The picture: three arrows point to a value; two point to "5", one points to "9". The voter picks "5" and throws away the outlier. One liar cannot outvote two truth-tellers.

Why the topic needs it: active redundancy (Triple Modular Redundancy) never switches anything off — it simply masks a wrong output by out-voting it. This is a different survival strategy from standby, and it connects to Byzantine Fault Tolerance when a faulty unit lies inconsistently, and to Single Event Upset (SEU) as a common cause of a single wrong bit.

Prerequisite map

Every arrow above is a "you must understand this before that". Trace any path and you can read the parent note top to bottom.

Related building blocks live in Power Budget (why hot standby's always-on cost matters), Failure Modes and Effects Analysis (FMEA) and Fault Detection and Isolation (how failure gets noticed before a switchover can happen). The parent topic is the redundancy note.

Equipment checklist

Cover the right side and answer each before opening the parent note.

What does $R (t)$ mean in plain words? ::: The probability a component is still working after running for time $t$ ; a number from $0$ to $1$ .
What range does the time variable $t$ take? ::: $t \geq 0$ — elapsed running time from launch, never negative.
What is $λ$ , precisely? ::: The instantaneous hazard rate — the fraction of currently surviving units that die per hour, not a plain count of failures per hour.
What does $d t$ mean? ::: An infinitesimally small slice of time (a duration shrunk toward zero).
What differential equation governs single-unit reliability, and what is its solution? ::: $d R / d t = - λ R$ , solved by $R (t) = e^{- λ t}$ .
What is $R (0)$ for any unit? ::: $1$ — at launch nothing has failed yet.
What is the chance a unit has failed by time $t$ ? ::: $1 - e^{- λ t}$ , the complement of the survival chance.
Why do we multiply $(1 - e^{- λ t})$ by itself for two units? ::: To get "both dead" — the multiply rule for independent events gives the overlap of both failing.
What does $\infty$ mean in $\int_{0}^{\infty}$ ? ::: Not a number, but "let $t$ grow forever"; safe because $e^{- λ t}$ decays to zero fast enough for a finite total.
What does $\int_{0}^{\infty} R (t) d t$ compute, and what does it look like? ::: The MTTF (average lifetime); geometrically it is the area under the survival curve.
What is MTTF for a single unit, and how is the integral done? ::: $1/ λ$ , from $\int_{0}^{\infty} e^{- λ t} d t = [- \frac{1}{λ} e^{- λ t}]_{0}^{\infty} = 1/ λ$ .
What does $(k n)$ mean and what is its formula? ::: The number of ways to choose $k$ of $n$ items; $(k n) = \frac{n !}{k ! ( n - k )!}$ .
What does majority( $A, B, C$ ) do? ::: Outputs whatever at least two of the three units agree on, ignoring the outlier.